Global well-posedness of solutions for the equations modelling the motion of a rigid body in a bidimensional perfect fluid

Abstract

This paper considers a system modelling the evolution of a rigid body immersed in a bidimensional incompressible perfect fluid. In the special case of a disk-shaped rigid body, it was shown by C. Rosier and L. Rosier (2009) that the system admits a unique global solution when the initial fluid velocity $u_0$ belongs to $H^s$ ($s \ge 3$) and its vorticity $\operatorname{curl} u_0$ lies in $L^p$ with $1 \le p < 2$. By establishing a Beale-Kato-Majda type bound, we generalize the result by removing the constraint $\operatorname{curl} u_0 \in L^p$ and allowing the rigid body to be of arbitrary shape. Moreover, we obtain an explicit energy bound.